we complete the construction, since the construction of the bit thread configurations in \(\mathcal {W}\left( R \right)\)  and \(\mathcal {W}\left( S \right)\)  are trivial (“coarse-grained"). In this way, we have reduced the original problem to a fairly simple subproblem involving only three elementary regions, with simple extra boundary conditions. By using the systematic method of convex dualization developed in [1]}, the existence of the solution to this convex optimization problem can be proved immediately. To be self-contained, we put the explicit proof in the appendix.
